Digit sums of multiples of 11

[I got this one from Madan Musuvathi.]

Some multiples of 11 have an even digit sum. For example, 7*11 = 77 and 7+7 = 14, which is even; 11*11 = 121 and 1+2+1 = 4, which is even. Do all multiples of 11 have an even digit sum? (Prove that they do or find the smallest that does not.)

February 2010

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